The role of ocean circulation in transforming surface forcing into interannual-to-multidecadal oceanic variability is an area of ongoing debate. Here, a novel method, establishing exact causal links, is used to quantitatively determine the role of ocean active and passive processes in transforming stochastic surface forcing into heat content variability. To this end, we use a global ocean model in which the dynamical response to forcing can be switched on (fully active) or off (purely passive) and consider the resulting effect on heat content variance. While the ocean passive processes mainly control the surface variance (over 92\%) in all basins, most regions show the importance of active processes at depth. This role is particularly important for full-depth North Atlantic heat content, which we investigate further, highlighting signatures of the meridional overturning circulation.